Wave Scattering by Twin Surface-Piercing Plates Over A Stepped Bottom: Trapped Wave Energy and Energy Loss

To evaluate the trapped wave energy and energy loss, the problem of wave scattering by twin fixed vertical surfacepiercing plates over a stepped bottom is numerically simulated using the open source package OpenFOAM and the associated toolbox waves2Foam. The volume of fluid (VOF) method was employed to capture the free surface in the time domain. The validation of the present numerical model was performed by comparing with both the analytical and experimental results. The effects of the spacing between two plates and the configuration of stepped bottom on the hydrodynamic characteristics, such as reflection and transmission coefficients, viscous dissipation ratio, and relative wave height between the plates (termed as trapped wave energy), were examined. Moreover, the nonlinear effects of the incident wave height on the hydrodynamic characteristics were addressed as well. The results show that the step configuration can be tuned for efficient-performance of wave damping, and the optimum configurations of the step length B, the step height h1 and the spacing b, separately equaling λ/4, 3h/4, and 0.05h (λ and h are the wavelength and the water depth, respectively), are recommended for the trapping of wave energy.


Introduction
The oil crisis of the 1970s opens up a new era of renewable energy exploitation (Vicinanza et al., 2014) and the most concerned areas include solar, wind and ocean energies (López et al., 2013). As an important renewable resource, wave energy has the basic characteristics of wide distribution, large energy density, simple collection structure and so on (Ning et al., 2014). Particularly, the oscillating water column (OWC) wave energy converter (WEC) is widely used for wave energy extraction because of its simple structure, easy maintenance and installation. However, due to the wave reflection from the front-wall and the vortex shedding in the vicinity of the lower tip of the frontwall, most incident wave energy is reflected and dissipated, which is one of the key reasons for the low conversion efficiency of the WECs. To investigate the complex hydrodynamic behaviors and further enhance the amount of wave en-ergy coming into the column, basically, the OWC device can be simply regarded as two truncated surface-piercing plates with an open top, which is also served as a breakwater system. Moreover, in the full-scale sea test, the bathymetry is arbitrary, which has a significant impact on the fluid dynamics.
In the literature, many scholars have devoted much effort to the interactions between ocean waves and thin-plate structures. As early as in the 1960s, Wiegel (1960) investigated the wave scattering by a single vertical plate, using both theoretical and experimental methods. He concluded that the transmitted wave energy is equal to the wave energy propagating through the vertical face just below the plate. Losada et al. (1993) employed the eigenfunction expansion method to study the oblique wave scattering by a vertical thin barrier, with the consideration of high-order harmonic waves near the barrier. Kriebel and Bollmann (1996) conducted physical experiments to predict the wave transmission past vertical barriers. They concluded that the modified power transmission theory provided better agreement with the data. Porter and Evans (1995) applied Galerkin method to investigate the single thin-plate problem and obtained similar conclusions with Losada et al. (1993). Moreover, the singular behaviors near the tip of plate were considered for a faster convergence in numerical computations. Zheng et al. (2006) investigated the wave scattering problem by a rectangular submerged breakwater parallel to a vertical plate. The wave loading on structures was examined.
As for the problem of interaction between regular waves and twin vertical plates, Ohkusu (1974) and Srokosz and Evans (1979) adopted theoretical methods and found that there existed resonance conditions at specific wave periods, which could lead to the full reflection and transmission phenomena on the hypothesis that if the interval between plates was large enough, and there would be little disturbance between each other. Stiassni et al. (1986) carried out experiments on the vertical plates and verified the resonance frequency proposed by Srokosz and Evans (1979), but unfortunately failed to well reveal the characteristics of the peak value. Apart from impermeable plate system, Isaacson et al. (1999) applied the eigenfunction expansion method to study the wave diffraction by two permeable vertical plates. Shin and Cho (2016) compared the results of theoretical and experimental methods, and explored the effects of plates spacing and draft on the reflection and transmission coefficients.
In recent years, many researchers have paid more attention to the influence of sea bottom profile instead of a flat one in coastal regions, which, in fact, is more suitable for the practical engineering application. Rezanejad et al. (2013) examined the role of stepped bottom topography in increasing the efficiency of an onshore OWC device, and the results showed that a proper configuration of sea bottom step significantly improved the capacity of power absorption. Ashlin et al. (2016) studied the effects of the bottom profile on an OWC device and the influence of the bottom configuration on the hydrodynamic performance was demonstrated. Ju et al. (2017) implemented both the theoretical and experimental techniques to evaluate the wave dissipation of the structure combined by one plate and one submerged breakwater, and how to obtain the minimum of transmission coefficient was discussed. Actually, due to the variation of sea bottom topography, whether the OWC device or the vertical plates system, the hydrodynamic performance may be entirely different compared with the one over a flat bottom.
In the present study, we utilized the open source software OpenFOAM to study the 2D numerical model of two vertical plates penetrating system. A rectangular submerged breakwater was introduced to simply represent the stepped sea bottom. The influence of the dimensions of the breakwater on hydrodynamic performances of the system was examined, including reflection and transmission coefficients, viscous dissipation ratio and relative fluctuation amplitude of free surface between plates, which may be used to quantify the trapped wave energy. Moreover, the effect of the wave nonlinearity has also been considered.

Governing equations
For incompressible fluid, the motion of the fluid with the assumption of laminar flow is governed by the continuity and momentum conservative equations, which can be described within the Cartesian reference frame, respectively, as: ∇ · U = 0; (1) U ρ g X where denotes the velocity vector, , the fluid density, p * , the pseudodynamic pressure, , the gravity acceleration, , the displacement vector, and μ eff , the efficient dynamic viscosity.
For the simulation of air-water two-phase flow problem, the function of volume fraction α, which is defined as the proportion of water volume in each cell, satisfies a classical transport equation:

∇
In OpenFOAM, an artificial compression term (Weller, 2002) ·U r α(1-α) has been added into Eq. (3) to diminish the smearing effects of the air-water interface and an especially designed useful solver named MULES (multidimensional universal limiter for explicit solution) (Deshpande et al., 2012) was applied to ensure the boundedness condition. Therefore, Eq. (3) can be rewritten as: where U r is the compressed velocity. It is noted that the newly introduced compression term works only at the air-water interface (Rusche, 2003). The discretization scheme and the detailed algorithm of solving the equations coupling the velocity and pressure fields can be found in Jasak (1996).

Method of generating and absorbing waves
The toolbox library named waves2Foam based on Open-FOAM was utilized to generate and absorb free surface waves simultaneously, as shown in Fig. 1. The solver inter-Foam, programmed in OpenFOAM to solve the air-water two-phase flow problem, was employed in this study. Detailed descriptions can be found in Jacobsen et al. (2012).
In this context, the linear wave theory was utilized, and the water particle velocities can be expressed as: where u x and u z are respectively the horizontal and vertical velocities of the water particle, H, the wave height of incident waves, ω, the angular frequency of the generated wave, k, the wave number, z, the vertical distance of the wave surface from the still water surface, and h, the water depth.

Model validation
3.1 Numerical convergence Being able to use less computational resources and timing-cost for the same order of accuracy, the numerical convergence verification is performed firstly. A preliminary 2D numerical wave tank without marine structures was constructed, and the essential parameters were as follows: the wave tank was 11λ (λ is the wavelength) in length and 0.8 m in height; the water depth was 0.4 m; the generated wave was 0.02 m in height and 0.8 s of period (the shortest wave tested in this paper). After convergence test, the grid number in the horizontal and vertical directions is 90 per wave length and 15 per wave height, respectively. The appropriate time step is 1/1000 of per wave period.
It is a common sense that for thin plates, simulation results are sensitive to the mesh applied, especially the mesh around those sharp tips. In Fig. 2, three different grids with coarse, medium and fine size around twin-plates were considered and their minimum grid sizes were respectively 0.001 m, 0.0008 m and 0.0005 m. For the three grids, the different levels of refinement within the plates were implemented in the horizontal direction. The smallest horizontal grid size was deployed on the inner sides of the plates (i.e., inside the column), and then became coarser gradually until equaling to the horizontal unrefined grid size. Additionally, the front of the plates, the area between the plates and the rear part of the plates were all put in monitoring wave gages to record the surface elevations at the same location for above grids with varied coarseness.
The monitored results, including the three grids simulated by laminar model (the efficient dynamic viscosity μ eff in Eq. (1) is 10 -3 Pa·s) and the medium grid using k-ω SST turbulence model, are displayed in Fig. 3. Except for the computed results between the plates a few differences exist, and there is almost no discrepancy for the other two places for different grids and different models, which indicates that the mesh convergence for thin plates has reached adequately. And moreover, the turbulence effects are actually minimal and ignorable, which is in accordance with the finding of Windt et al. (2018).
Thus, for further investigation, laminar model is employed completely.

√ −1
To validate the numerical model, a semi-analytical theory of wave scattering by twin surface-piercing plates over a rectangular submerged breakwater was developed under the framework of linear water wave theory. For two-dimensional problem, there exists a velocity potential Ф (x, z, t), which satisfies the Laplace equation together with the free surface boundary, sea bottom boundary, wetted solid body boundary and Sommerfeld radiation conditions. Based on the matched eigenfunction expansion method, the complex amplitude of velocity potential ϕ (Ф (x, z, t)=Re{ϕ(x, z) e -iωt }, where i = ) can be divided into four notations, namely,  By separation of variables, the formal expressions of velocity potentials in different sub-regions may be written as: where A is the incident wave amplitude, A 0 , A m , B n , C n , D n , E n and F m are unknown coefficients to be determined. In addition, following Mei and Black (1969), the vertical eigenfunctions can be expressed as: k Note here that the wave numbers k m and are calculated by the following equations (the positive real roots to be taken), respectively, { By the matched eigenfunction expansion method, the continuities of the horizontal velocity and pressure across the common boundary are usually employed to guarantee the connectivity of flow field and to determine the unknown coefficients in the formal expressions of velocity potentials. It is worth mentioning that, to speed up the convergence and improve the accuracy in numerical computations, the singular behaviors in the vicinity of sharp corner have been characterized by orthogonal polynomials (Porter and Evans, 1995;Deng et al., 2013).
As the main purpose of this study is to simulate the trapped wave energy (traveling into the column) and energy loss induced by the vortex shedding, the detailed solving procedure has been omitted for brevity. Those who are interested in the detailed derivations can refer to Deng et al. (2013). Therefore, once the values of unknowns A 0 , A m , B n , C n , D n , E n and F m are obtained, the reflection C r and transmission C t coefficients, characterized by the outgoing waves  DENG Zheng-zhi et al. China Ocean Eng., 2019, Vol. 33, No. 4, P. 398-411 401 at infinity, can be expressed as: There are several infinite summations, such as and , in the velocity potential expansions and the expressions of the horizontal velocity distribution functions on the common interfaces. The truncation of these summations determines the accuracy in the numerical computations. In this study, to obtain three decimal places, the truncation value for both the m-and n-summations is 400. For other summations, appropriate truncation values are selected by convergence verification (for more details, readers can refer to the work by Deng et al. (2013)).
In order to verify the theoretical model and the code to calculate the variations of reflection and transmission coefficients against the dimensionless circular frequency ω 2 h/g after the convergence verification, Fig. 5 displays the comparisons of C r and C t for the present limited case with the work by Porter and Evans (1995). It is obvious that a good agreement is achieved.
With the above verifications, the well-validated semianalytical model can be used to check the current CFD model. In Fig. 6, the results of reflection and transmission coefficients were compared with each other under the conditions of B/h=1, h 1 /h=0.5, b/h=0.05, and four different wave heights (i.e., H=0.005, 0.01, 0.015 and 0.02 m). Generally speaking, except the discrepancy induced by the viscosity and vortex shedding (especially in the vicinity of the lower edge of front plate) being ignored in the above theoretical deduction, a similar trend for both the reflection and transmission coefficients can be observed, which further illustrates the correctness and validity of the numerical model. Moreover, it is worth noting that a higher wave height can contribute to a smaller transmission coefficient due to the fact that the high-order harmonic waves are subjected to the obstruction by the two surface-piercing plates system, which will be examined and discussed in detail in the following sections.

Verification of the surface elevations between the plates
In order to further validate the capability of capturing the free surface between the vertical plates, the problem of wave scattering by two surface-piercing plates over a flat bottom, investigated by Shin and Cho (2016), was studied subsequently. Same as the experimental setup by Shin and Cho (2016), the wave tank is 20 m long and 0.4 m deep, and the target wave height is 0.02 m. What we were interested in was the oscillating amplitude of the free surface between the plates, and present numerical results were compared with the theoretical and numerical results conducted by Shin and Cho (2016). Two wave gages were put in the same locations with Shin and Cho between the plates to monitor the surface elevations and the compared results are shown in Figs. 7a and 7b. The abscissa f represents the reciprocal of the period T of the incident wave, and the ordinate A/A i is the ratio of the free surface wave amplitude between the plates to the incident wave amplitude.
In general, the present numerical results achieve a good agreement with both analytical and experimental ones (Shin and Cho, 2016), except for the vicinity of natural resonance frequency, which causes significant nonlinear effects. And the assumptions of small amplitudes become erroneous when the incident wave frequency is close to the resonance frequency (Howe and Nader, 2017), which is another key factor that results in discrepancies arise. Anyway, this agreement has again confirmed the correctness of the present numerical model in simulating the wave-body interaction problems.

Results and discussion
In the rest of this paper, the impacts of the submerged step bottom on the twin plates system will be simulated and analyzed. Physically, due to the existence of a stepped bottom, which essentially changes the water depth, the wave deformation will occur. Further, the hydrodynamic performances could be affected inevitably. Thus, the effects of the stepped bottom on the reflection (C r ) and transmission (C t ) coefficients as well as the wave energy dissipation ratio (E Dis ) and the relative wave height (η) between the plates will be examined.
By referring to Fig. 1, the initial incident wave height is set as 0.02 m in this paper. The numerical wave tank is of the length L=20 m and the water depth h=0.4 m. As for the two vertical plates, the right plate is located 12.09 m away  (Porter and Evans, 1995). from the wave-maker boundary. The location of the left plate is determined by the spacing interval b. Actually only four plates spacings, i.e., b/h=0.05, 0.1, 0.15, 0.2, were taken into account. And the immersed depth of both plates was d=0.05 m (d/h=1/8) for all the coming simulations.
To investigate the wave deformation, nine numerical wave gages (G 1 -G 9 ) are positioned to monitor the free surface displacements as shown in Fig. 1. The locations of the nine numerical gages are listed in Table 1. Among these gages, G 1 -G 3 are allocated in front of the step, G 4 -G 5 above the step, G 6 -G 8 in between the twin plates, and G 9 behind the step. It is noted that the time series captured by G 1 -G 5 are used to separate the incident and reflected waves through the two-point method proposed by Goda and Suzuki (1977). G 6 -G 8 are used to monitor the fluctuation of the free surface in between the two plates and their locations vary with the spacing interval b as shown in Table 1. The time series captured by G 9 is used to calculate the transmission coefficient C t , which is the ratio of the transmitted wave height to the incident one.

Variation of the relative step length B/h
In this subsection, the effects of the relative step length B/h on the hydrodynamic performances are examined first. Without loss of generality, a single wave period (T=1 s) and four different spacing intervals (i.e., b/h=0.05, 0.1, 0.15, 0.2) were chosen in the following computations. The step height was fixed to be 0.2 m (h 1 /h=0.5). And by referring to Fig. 1, eight relative step lengths, i.e., B/h=0, 0.2, 0.4, 0.6, 0.8, 1, 1.5, 2, were considered. It is worth mentioning that the cubic spline curve fitting is used in the following figures to characterize the variation trend of the interesting quantities. Although the smoothness of curves has been artificially improved to a certain extent, the appearance of peak and trough values is not affected (similar with the treatment by Ning et al. (2015)).

Effects on the reflection and transmission coefficients
In Fig. 8a, the variation of reflection coefficients C r in front of the system, depicted in Fig. 1, against the relative step width B/h are given. Generally speaking, the curves do     (Rezanejad et al., 2013). However, the varying processes of the cases of four spacing intervals b/h are almost consistent, except for the slight phase difference of peak. It is also noticeable that for all the cases of B/h > 0, the calculated reflection coefficients C r are all larger than those for B/h =0, which reveals that due to the existence of a step, the capability of wave damping is improved to some extent. Additionally, broadening the width between the two plates can reflect more wave energy. Fig. 8b shows the variations of transmission coefficients C t against B/h for four different widths between the two plates. As expected, the transmission coefficient C t exhibits a completely opposite trend compared with C r in Fig. 8a. Moreover, in the absence of a step (i.e., B/h=0), the values of transmission coefficient are maximum for all cases, which indicates that the submerged step strengthens the turbulence and vortex shedding, and further improves the performance of wave damping. It is straightforward to conclude that the relatively wider spacing between the two plates can block more wave energy, and the optimum step length, as a breakwater, is in the range of B/h =0.2-0.4.

Effects on viscous energy dissipation
As mentioned previously, the penetrating system involving twin vertical plates can be simply regarded as a fundamental structure of a fixed OWC device with the top opening. For an OWC WEC device, the most concern is usually the amount of energy absorption from incident waves, which is in turn determined by the available energy trapped by the double plates. To allow more wave energy coming into the region bounded by the two plates, it is necessary to assess the viscous energy dissipation due to turbulence and vortex shedding, and seek for an optimum structure configuration with less energy dissipation. According to the energy conservation principle, the total incident wave energy should be equal to the sum of the reflected, absorbed, transmitted, and dissipated energies. Thus we have the following relationship: In this paper, the power take-off model was not added, such as air turbine, power generator set and so on. Therefore, the absorbed energy in Eq. (14) is E Abs =0. Meanwhile, given the fact that the magnitude of wave energy is directly proportional to the square of wave height (H 2 ), the quantities of E Ref and E Tran can be quantified by the square of reflection coefficient ( ) and those of transmission coefficient ( ), respectively. In this way, the value of E Dis may be estimated by the expression of 1-( + ), which is also called the energy dissipation ratio in this paper. Fig. 9 shows the variation of wave energy dissipation ratio as a function of B/h under the conditions of four horizontal spacings (i.e., b/h=0.05, 0.1, 0.15, 0.2). Physically, as the length of the submerged step becomes larger and the interacting time with waves prolongs enough, the wave energy dissipation ratio will increase significantly, and the occurrence probability of factors causing wave energy loss, such as wave breaking, vortex structure formation and shedding, will be further intensified. This rule is dressed in Fig.  9 when the width B varies from 0 to h, where the dissipation ratio varies from the minimum to maximum values. Rezanejad et al. (2013) reported that there existed two resonance mechanisms: (1) the fluid volume above the step and outside the left plate; (2) the fluid volume enclosed by the double plates. It is obvious that the peaks of energy dissipation are induced by the well-known quarter wavelength resonance in a half open organ pipe. Since, when B/h equals 1, the spacing B is approximately to the multiple of λ/4 for T=1 s.

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DENG Zheng-zhi et al. China Ocean Eng., 2019, Vol. 33, No. 4, P. 398-411 Moreover, the influence of the horizontal spacing b on the energy dissipation ratio is also displayed in Fig. 9. It is clear that the relatively wider the spacing, the more the dissipation ratio, and the differences between the maximum and minimum energy dissipation values for b/h=0.05, 0.1, 0.15 and 0.2 are 2.306%, 3.636%, 5.041% and 6.501%, respectively.

Effects on trapped wave energy by plates
In view of the working principle of the OWC device, the spatially averaged free surface displacement is usually used to quantify the trapped wave energy by plates, which directly determines the performance of the device. According to Wang et al. (2018), the extraction efficiency is proportional to the dimensionless surface elevation and air pressure in the chamber. Therefore, a relative wave height η, calculated by a ratio of the averaged surface displacement to the incident wave height, is introduced to estimate the effects of the step length on the trapped wave energy. Generally, the higher the relative wave height, the more the trapped energy, especially, as the incident wave frequency approaches or equals the resonance one of the enclosed OWC.
Basically, when the breadth between the twin plates is relatively small, the inside free surface moves like a weightless piston, and then the wave elevation at the central location can represent the whole vertical displacement of the interior free surface (Ning et al., 2016). Fig. 10 displays the measured data by G 6 -G 8 , which confirms that it is reasonable to adopt the surface elevation at the middle place, or more precisely, the averaged value of three points in between the plates to represent the whole vertical motion. It is noted that, in this paper, the averaged wave height is evaluated by the mean value of the wave heights monitored by G 6 -G 8 at a stable stage.
The variation of the relative wave height against the dimensionless step length B/h is shown in Fig. 11. It is found that the minimum of η can be only achieved in the absence of the step bottom. With the presence of the step, the water depth condition has been changed. As a result, the so-called 'shoaling effect' happens. And this is why the relative wave height is always larger than 1. Additionally, as the step length increases gradually, the relative height raises up correspondingly. When the length equals λ/4, the first peak of η occurs, and then falls down until the next resonance phenomenon arrives. It is noted that the first peak occurs at about B/h=1.1. Thus, in the present dimensionless range of interest (B/h), the second peak of η exceeds the range and cannot be displayed in Fig. 11.
It is also seen that the minimum spacing (i.e., b/h=0.05) can offer a relatively more drastic surface fluctuation, however, the wave energy trapped in between the plates is limited. Nevertheless, with the construction cost and the satisfactory extraction efficiency, the configuration of b/h=0.05 and B=λ/4 is recommended for the design of the OWC WEC device.

Variation of relative step height h 1 /h
The effects of the variation of step height h 1 are discussed in this subsection under the conditions of T=1.0 s, H=0.02 m, and d/h=1/8. Owing to the relatively small wave height, three step heights, i.e., h 1 /h = 4/8, 5/8, and 6/8, are chosen, as listed in Table 2.

Effects on the reflection and transmission coefficients
The variation of the reflection coefficient C r against B/h is plotted in Fig. 12. It is found that for different spacings, the curves of C r have the almost identical trend corresponding to the individual step height. Moreover, the increase of the step height needs a shorter step length to achieve the resonance condition. As expected, the higher the step height, the more the reflected wave energy, which can be considered as a vertical wall with a gap.   Fig. 11. Relative wave height between plates relative step versus the relative step width B/h. Fig. 13 shows the transmission coefficient C t as a function of dimensionless step length B/h. It can be clearly seen that the step height has a significant influence on C t . The larger the relative height h 1 /h, the smaller the transmission coefficient, which indicates that more wave energy has been damped before coming into the rear of the system. Similar to the reflection coefficient C r , due to the varied resonance conditions, the transmission coefficient C t reaches a peak value when B/h are approximate to 1, 0.8, 0.6 for h 1 /h=1/2, 5/8, 3/4, respectively.
In general, when the twin plates system is designed as a breakwater, the higher step height is more conducive to its  special function of wave damping. However, when serving as an OWC WEC device, a step height optimized for allowing more wave energy to come into the column is necessary and preferable. Fig. 14 shows the variations of the wave energy dissipation with respect to B/h. Evidently, a higher submerged ob-ject causes more energy dissipation due to the complex fluid motions in the vicinities of the lower tips of the plates and upper corners of the step. It is well-known that vortex generation and shedding is an important factor which will result in considerable wave energy dissipation. As shown in Fig. 15, the streamline structures for the three step heights are illustrated on the basis of four different time instants. It can be found that at the same time, the higher step height  DENG Zheng-zhi et al. China Ocean Eng., 2019, Vol. 33, No. 4, P. 398-411 407 will induce the generation and shedding of more and larger vortices in the vicinities of the lower edges of twin plates and the upper corners of the step. Particularly, when the resonance phenomenon happens, more energy will be dissipated due to the strongly nonlinear flow.

Effects on the wave fluctuation between plates
There is no doubt that the existence of submerged dike will change the water depth condition when the wave is propagating to shoreline. The most direct influence on waves is the variation of wave height. The level of the deformed wave height due to the shoaling effect is mainly determined by the deformation coefficient k s of the shallow water, which is calculated by According to Eq. (15), it is easy to find that with the decrease of water depth h s , the shoaling coefficient k s will increase and then the shallow-water effect is more evident. In this paper, the corresponding shoaling coefficients k s for the three step heights are 1. 0848, 1.2009, and 1.4221, respectively, which indicates that the wave height over the step has been amplified to some extent.
The variations of the relative wave height with respect to B/h are shown in Fig. 16. It can be found that the greater the height of submerged step, the larger the amplitude of wave elevation in the middle of two plates, which is in accordance with what we have expected for all step heights. With the increase of the spacing, the relative wave height of the same step configuration reduces correspondingly. To trap more wave energy, the configuration of h 1 /h=6/8 is re-commended.

Nonlinear effects on two vertical system
Based on the linear wave theory, there is no direct relation between the wave energy conversion efficiency of OWC device and the incident wave height. However, according to the works by Luo et al. (2014), Elhanafi et al. (2016) and Ning et al. (2015), the enhanced nonlinear effects due to an increased wave height will seriously affect the working efficiency of OWC devices. In this paper, four different incident wave heights, i.e., H=0.01, 0.02, 0.03, and 0.04 m, are firstly chosen to investigate how the wave nonlinearity (wave steepness H/λ) influences the hydrodynamic performances of the penetrating system.
In Wang et al. (2018), the fast Fourier transform (FFT) algorithm was employed to analyze the high-order components of the time series of the surface elevation sampled by numerical wave gages, which were separately allocated in Region I (over the step and outside the plates), Region II (in between the twin plates), and Region III (behind the plates), as shown in Fig. 1. Fig. 17 shows the FFT results for B/h=1, h 1 /h=1/2, d/h=1/8, and T=1 s. Remarkably, the spacing b/h=0.05 is taken into account owing to its more serious interior surface fluctuation displayed in Fig. 11. With the enhancement of wave steepness, the fundamental wave components coincide well with each other, while the second-order harmonic waves become more and more significant, as shown in Fig. 17b. It is also seen that the proportion of second order harmonic components inside the plates is much more than that in Region I. And Fig. 17c shows that the high-order harmonic waves are negligible for the locations at the back of the two plates. According to Wang et al. (2018), the high- Fig. 16. Relative wave height versus B/h for different relative step heights h 1 /h. order harmonic components are hard to transmit into the interior of the OWC. In other words, the high order harmonic waves inside the column are much less than those outside the plates. However, the contradiction in this study is due to the occurrence of resonance phenomenon when the incident wave period is 1 s and B/h is nearly to 1, which leads to a more drastic surface oscillation and triggering more high-order waves to generate.
Apart from the above condition of T=1.0 s, a series of tests, for T=0.8, 0.9, 1.1, 1.2, and 1.3 s, were carried out as well, and the results of the least and largest wave periods, T=0.8 s and T=1.3 s were illustrated in Fig. 18. It can be observed from Fig. 18a, the second-order harmonic waves in Region I are larger than those in Region II, even though the high-order waves are quite smaller than the fundamental waves. Similar results can be found in Fig. 18b. Therefore, it can be concluded that except for the occurrence of resonance condition (T=1 s), the internal high-order wave components are less than the external, which coincides with the findings by Wang et al. (2018). Besides, it is worth mentioning that for all the cases tested, all the high-order components in Region III can be ignored due to the weak transmission capability. However, for the components in Regions I and II, with the increase of wave steepness, more fundamental wave energy is transformed into the high-order ones.
Moreover, the variation of wave nonlinearity against B/h were investigated, and five wave heights (i.e., H=0.005, 0.01, 0.02, 0.04, 0.08 m, and the corresponding wave steepness H/λ = 0.0034, 0.0068, 0.0137, 0.0273, 0.0547) were considered, ranging from weak nonlinearity to relatively strong nonlinearity.  Fig. 18. FFT results for different incident wave periods. DENG Zheng-zhi et al. China Ocean Eng., 2019, Vol. 33, No. 4, P. 398-411 Given a single structure configuration of h 1 /h=0.5, b/h= 0.05 and under the excitation of regular waves of period T=1 s, the variations of the reflection and transmission coefficients, energy dissipation ratio, and relative wave height with respect to B/h are depicted in Figs. 19-21. As shown in Figs. 19a and 19b, the similar sinusoidal curves can be observed. It is particularly noted that as the incident wave hei-ght increases, both the reflection and transmission coefficients decrease for wave steepness H/λ=0.0034-0.0137. Nevertheless, when the wave steepness increases and nonlinearity enhances, the reflection coefficients will be larger than the conditions with smaller wave steepness for a longer B/h. The energy dissipation ratio increases with the increasing incident wave height, as shown in Fig. 20, which indicates that the stronger the nonlinear effect, the greater the energy loss.
In Fig. 21, the relative wave height between the plates against B/h for different incident wave heights is illustrated. It can be seen that increasing the incident wave height, when the wave steepness H/λ is smaller than 0.0137, is beneficial to trap more wave energy traveling into the column bounded by plates, but to reduce the energy flux. In contrast, for H/λ=0.0273 and 0.0547, the wave height between the plates is inferior to the incident wave, due to the conversion of prime frequency waves to higher-order harmonic waves, and the weak transmissive ability of harmonics constrains the propagation into the interior of the plates. Consequently, the relative wave height will be limited.

Conclusions
By employing the open source package OpenFOAM and the toolbox waves2Foam, the wave scattering by a twinplates system over a submerged step was numerically in-vestigated. Compared with the analytical and experimental results, the current numerical model was well validated. Through varying the dimensions of step, the hydrodynamic characteristics, such as the reflection and transmission coefficients, energy dissipation ratio, and relative wave height, were examined and analyzed. In addition, by increasing the incident wave height, the effects of wave nonlinearity on the hydrodynamic performances of the system were demonstrated as well. From the numerical results and discussions, the following conclusions can be drawn.
(1) To some extent, the existence of a submerged step can enhance the wave damping of the twin-plates system, serving as a breakwater. Generally, the larger the step length is, the more the viscous dissipation will be. Besides, when the step length B equals an integral multiple of λ/4, the resonance phenomenon over the step and outside the plates happens, which results in the most drastic surface fluctuation inside the column. In this way, the amount of the trapped wave energy is significantly improved.

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DENG Zheng-zhi et al. China Ocean Eng., 2019, Vol. 33, No. 4, P. 398-411 (2) Increasing the step height leads to the decrease of the transmission coefficient and the increase of the reflection coefficient, which is beneficial for wave damping. The wave energy dissipation ratio enhances due to the vortex generation and shedding in the vicinities of the lower tips of plates and the upper corners of step. Besides, the increase of the step height seems to block more wave energy coming into the column bounded by the twin plates, however, it is helpful for the enhancement of the inside relative wave height, especially for the resonance situation. In this study, the step height of h 1 /h=3/4 is recommended.
(3) The enhancement of wave steepness will result in more portions of primary wave energy to be transformed into high-order components, which reduces the amount of wave energy propagating into the back of the system and strengthens the viscous dissipation. Additionally, increasing the wave steepness in a particular range is beneficial for the trapping of more wave energy in between the twin plates, but decreases the power flux ratio compared with the incident one. A relative strong nonlinearity, i.e., H/λ=0.0547, is of no effectiveness for wave trapping as the high percentage of wave energy loss.